MGF Technique

<Moment Generating Function Technique> 

We can define the relationship of distributions and the property of distribution by using properties of Moment generating functions. 

The main reason which makes it possible is that the moment generating functions are unique for each distribution. 

So If we know the moment generating function, there should be only distribution which correspond to the MGF. 

Let me give you some typical relationships and additive properties of some famous distributions and prove it by showing unique MGFs. 

<Distributions that have properties of additivity>

Binomial 

Chi-Square

Poisson

Gamma 

<Proof for Central limit theorem by using MGF >

Proof for Standardized mean --> Central limit theorem

Proof for General sums --> Central limit theorem

<Relations of some distributions using MGF Technique>

Standard Normal distribution with Chi-Square distribution

Gamma distribution with Chi-Square distribution

Exponential distribution with Gamma distribution

<Normal Approximation to Binomial>

Proof with Logarithm

I want to show that for some times, using Logarithm can make some proof more easier. The reason is that MGF function which is E(e^tx) is the e function so for some case, delete e by adding log makes simple formation. However, you should notice that for the case you add temporary logarithm, You should not forget to add E at the and. 

  

<Poisson Approximation to Binomial>